Problem 20 - Calculus Early Transcendentals

Question

a. Find the nth-order Taylor polynomials of the given function centered at \(0,\) for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=(1+x)^{-2}$$

Step-by-Step Solution

Answer

Question: Find the Taylor polynomials of order 0, 1, and 2 for the function \((1+x)^{-2}\) centered at x=0 and explain their graphical representation. Answer: The Taylor polynomials of order 0, 1, and 2 are: - 0th-order: \(P_0(x) = 1\) - 1st-order: \(P_1(x) = 1 - 2x\) - 2nd-order: \(P_2(x) = 1 - 2x + 3x^2\) Graphically, these polynomials provide local approximations of the original function \((1+x)^{-2}\) around the point x=0. The 0th-order polynomial is a horizontal line at y=1, the 1st-order polynomial is a decreasing line, and the 2nd-order polynomial is a parabola. As the order of the polynomial increases, the accuracy of the approximation also increases.

Step 1: Find Taylor Polynomial of Order 0

To find the zeroth-order Taylor polynomial, we evaluate the function for the point x=0, giving \(f(0) = (1+0)^{-2} = 1\). The zeroth-order Taylor polynomial is a constant function and matches the function's value at x=0, so \(P_0(x)=1\).

Step 2: Find Taylor Polynomial of Order 1

To find the first-order Taylor polynomial, we first calculate the first derivative of the function: \(f'(x) = \frac{-2}{(1+x)^{3}}\). Now, we evaluate the derivative for the point x=0: \(f'(0) = \frac{-2}{(1+0)^{3}} = -2\). The first-order Taylor polynomial is obtained by adding the first derivative to the zeroth-order polynomial: \(P_1(x) = 1 - 2x\).

Step 3: Find Taylor Polynomial of Order 2

For the second-order Taylor polynomial, we need the second derivative of the function: \(f''(x) = \frac{6}{(1+x)^{4}}\). Then we evaluate this second derivative for the point x=0: \(f''(0) = \frac{6}{(1+0)^{4}} = 6\). The second-order Taylor polynomial includes the first and second derivatives: \(P_2(x) = 1 - 2x + 3x^2\).

Step 4: Graph the Taylor Polynomials and the Function

To graph the original function and the Taylor polynomials, we can use software such as Desmos or GeoGebra or we can make a rough sketch on graph paper. The function graph will have a vertical asymptote at x=-1 and will approach infinity as x decreases. The Taylor polynomials will approximate the function in the local vicinity of x=0. As we go higher in polynomial order, the accuracy of the approximation increases. Comparing the Taylor polynomials and the original function, we should see that: - The 0th-order polynomial (\(P_0(x)=1\)) will just be a horizontal line crossing the y-axis at y=1. - The 1st-order polynomial (\(P_1(x) = 1 - 2x\)) will be a decreasing line. - The 2nd-order polynomial (\(P_2(x) = 1 - 2x + 3x^2\)) will be a parabola. These graphs will show how the Taylor polynomials provide local approximations of the original function \((1+x)^{-2}\) around the point x=0.